A general framework for prediction in penalized regression

There are two main approaches to carrying out prediction in the context of penalized regression: with low-rank basis and penalties or through the smooth mixed models. In this article, we give further insight in the case of P-splines showing the influence of the penalty on the prediction. In the context of mixed models, we can connect the new predicted values to the observed values through a joint normal distribution, which allows us to compute prediction intervals. In this work, we propose an alternative approach, called the extended mixed model approach, that allows us to fit and predict data simultaneously. The methodology is illustrated with two real datasets, one of them on aboveground biomass and the other on monthly sulphur dioxide ( SO 2 ) levels in a selection of monitoring sites in Europe.